Theorem lvecring

Description: The scalar component of a vector space is a ring. (Contributed by SN, 28-May-2023)

	Ref	Expression
Hypothesis	lvecring.1	$$⊢F=\mathrm{Scalar}\left(W\right)$$
Assertion	lvecring	$$⊢W\in \mathrm{LVec}\to F\in \mathrm{Ring}$$

Proof

Step	Hyp	Ref	Expression
1		lvecring.1	$$⊢F=\mathrm{Scalar}\left(W\right)$$
2		lveclmod	$$⊢W\in \mathrm{LVec}\to W\in \mathrm{LMod}$$
3	1	lmodring	$$⊢W\in \mathrm{LMod}\to F\in \mathrm{Ring}$$
4	2 3	syl	$$⊢W\in \mathrm{LVec}\to F\in \mathrm{Ring}$$